Supplementary Material for The Cusp Catastrophe Model as Cross-Sectional and Longitudinal Mixture Structural Equation Models

Supplementary Material for The Cusp Catastrophe Model as Cross-Sectional and Longitudinal Mixture Structural Equation Models Sy-Miin Chow Pennsylvania State University Katie Witkiewitz University of New Mexico Raoul Grasman University of Amsterdam Stephen A. Maisto Syracuse University This document features supplementary material from the Chow et al. s the paper entitled The Cusp Catastrophe Model as Cross-Sectional and Longitudinal Mixture Structural Equation Models. 0.1 Mplus Scripts for Fitting MSEM Model 5 TITLE: MSEM Model 5, cusp-inspired mixture model, unknown beta group DATA: FILE = thedat01.csv ; VARIABLE: NAMES = id alpha beta alphabeta y ybeta1 ybeta2 ybeta3 betacat; MISSING = ALL (-999); USEV = alpha ybeta1-ybeta3 y; classes = c0(2) c1(3); ANALYSIS: TYPE = MIXTURE; STARTS = 50 5;STSCALE=1.5; STITERATIONS = 100; MODEL: 1

!-------------------------------------------------- %OVERALL%!-------------------------------------------------- c1 on alpha@0; c1 on beta@0; ly by y@1; [y@0]; y@0; [c1#1@-10];! (lowbtoh); [c1#2@-10];! (lowbtol); c1#1 ON c0#1*15 (hibtoh); c1#2 ON c0#1*15 (hibtol); beta by ybeta1@1; beta by ybeta2*1 (l1); beta by ybeta3*1 (l2); [ybeta1-ybeta3@0];!-------------------------------------------------- Model c0:!-------------------------------------------------- %c0#1% [beta*2] (mbetah); beta*.5 (vb1); c1#1 ON alpha*3 (inita); c1#2 ON alpha*-3 (inita2); %c0#2% [beta*-2] (mbetal); beta*.5 (vb2); c1#1 ON alpha@0; c1#2 ON alpha@0;!-------------------------------------------------- Model c1:!-------------------------------------------------- %c1#1%!high beta, high DV ly on alpha*.5 (balpha1); [ly*2] (inth); ly*.1 (vqhighb); ly ON beta*.1 (bbeta2); %c1#2%!high beta, low DV ly on alpha*.5 (balpha1); 2

[ly*-2] (intl); ly*.1 (vqhighb); ly ON beta*-.1 (bbeta3); %c1#3%!low beta, med DV ly on alpha*.3 (balpha2); [ly*.01] (intm); ly*1 (vqlowb); ly ON beta@0;!-------------------------------------------------- MODEL CONSTRAINT:!-------------------------------------------------- New (lowbtoh lowbtol PHIBTOH PHIBTOL PHIBTOM PLOWBTOH PLOWBTOL PLOWBTOM); lowbtoh = -10; lowbtol = -10; PHIBTOH = exp(lowbtoh + hibtoh)/ (exp(lowbtoh + hibtoh) + exp(lowbtol +hibtol) + exp(0)); PHIBTOL = exp(lowbtol +hibtol)/ (exp(lowbtoh + hibtoh) + exp(lowbtol +hibtol) + exp(0)); PHIBTOM = exp(0)/ (exp(lowbtoh + hibtoh) + exp(lowbtol +hibtol) + exp(0)); PLOWBTOH = exp(lowbtoh)/ (exp(lowbtoh) + exp(lowbtol) + exp(0)); PLOWBTOL = exp(lowbtol)/ (exp(lowbtoh) + exp(lowbtol) + exp(0)); PLOWBTOM = exp(0)/ (exp(lowbtoh) + exp(lowbtol) + exp(0)); inth > intm; intm > intl;!-------------------------------------------------- OUTPUT: TECH1 TECH4 CINTERVAL; 3

SAVEDATA: RESULTS = 01outPar94.txt ; FILE = 01Cross94.class ; RECORDLENGTH IS 5000; SAVE = CPROBABILITIES; 0.2 Mplus Scripts for Fitting MSEM-RS Model 5 TITLE: T > 1 cusp-inspired mixture model, unknown beta group!#------------------------------------------------------------!# c1!#------------------------------------------------------------!# c0#1 #2!#-----------------------------------------------------------!# Hi Low!#High beta lowbtom + hibtoh*+inita*alpha 0!# Med Med!#Low beta lowbtom = [c1#1@10] 0!#-------!#Note: inita = c1#1 on alpha1 in c0#1!@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@!#conditional transition log-odds!#-------------------------------------------------------!#c0 = 2 (Low beta): reference group!#--------------------------------------------------------!# c2!# --------------------------------------------------!# c1 Med Med!#-------------------------------------------------------!#Med MtoOth + lmtom (c2#1 ON c1#1@20) 0!#Med MtoOthM = [c2#1@-10] 0!#-------------------------------------------------------!#!#c0 = 1 (High beta)!#---------------------------------------------------------!# c2!# --------------------------------------------------!# c1 High Low!#---------------------------------------------------------!#High MtoOthM + lowtoh + lhtoh + lot11a*alpha 0!#Low MtoOthM + lowtoh + lot21a*alpha 0!#---------------------------------------------------------!#Note: lot11a e.g., = c2#1 ON alpha2 in c0#1.c1#1!#lot21a e.g., = c2#1 ON alpha2 in c0#1.c1#2 4

DATA: FILE = thedatlong1.csv ; VARIABLE: NAMES = id alpha1 alpha2 alpha3 alpha4 alpha5 alpha6 betao ab1 ab2 ab3 ab4 ab5 ab6 y1 y2 y3 y4 y5 y6 ybeta1 ybeta2 ybeta3 betacat; MISSING = ALL (-999); USEV = y1-y6 alpha1-alpha6 ybeta1-ybeta3; classes = c0(2) c1(2) c2(2) c3(2) c4(2) c5(2) c6(2); ANALYSIS: TYPE = MIXTURE; STARTS = 80 5; STSCALE= 2;!STITERATIONS = 100; MODEL:!----------------------------------------- %OVERALL%!----------------------------------------- c1 on alpha1@0; c2 on alpha2@0; c3 on alpha3@0; c4 on alpha4@0; c5 on alpha5@0; c6 on alpha6@0; c1 on beta@0; c2 on beta@0; c3 on beta@0; c4 on beta@0; c5 on beta@0; c6 on beta@0; ly1 by y1@1; ly2 by y2@1; ly3 by y3@1; ly4 by y4@1; ly5 by y5@1; ly6 by y6@1; [y1-y6@0]; y1-y6@0; ly1 with ly2@0; ly1 with ly3@0; ly1 with ly4@0; ly1 with ly5@0; ly1 with ly6@0; 5

ly2 with ly3@0; ly2 with ly4@0; ly2 with ly5@0; ly2 with ly6@0; ly3 with ly4@0; ly3 with ly5@0; ly3 with ly6@0; ly4 with ly5@0; ly4 with ly6@0; ly5 with ly6@0; [c1#1@10];! (lowbtom); [c2#1@-10];!(mtoothm) [c3#1@-10];!(mtoothm) [c4#1@-10];!(mtoothm) [c5#1@-10];!(mtoothm) [c6#1@-10];!(mtoothm)! Notice that LO(low to high high beta) = (MtoOthM + lowtoh) c1#1 ON c0#1*-10 (hibtoh);!expecting to be around 10 LO(low to high high beta)!as P(lo to high high beta) when alpha = 0 should be close to.5. c2#1 ON c0#1*11 (lowtoh);!effect of c0#1 on [c2#1] c3#1 ON c0#1*11 (lowtoh); c4#1 ON c0#1*11 (lowtoh); c5#1 ON c0#1*11 (lowtoh); c6#1 ON c0#1*11 (lowtoh); beta by ybeta1@1; beta by ybeta2*1 (l1); beta by ybeta3*1 (l2); [ybeta1-ybeta3@0];!----------------------------------------- Model c0:!----------------------------------------- %c0#1% [beta*2] (mbetah); beta*.5 (vb1); ly1 on alpha1 (balpha1); 6

ly2 on alpha2 (balpha1); ly3 on alpha3 (balpha1); ly4 on alpha4 (balpha1); ly5 on alpha5 (balpha1); ly6 on alpha6 (balpha1); c1#1 on alpha1 (inita);! Notice that LO(high to high high beta) = (MtoOthM + lowtoh + lhtoh) c2#1 ON c1#1*.1 (lhtoh); c3#1 ON c2#1*.1 (lhtoh); c4#1 ON c3#1*.1 (lhtoh); c5#1 ON c4#1*.1 (lhtoh); c6#1 ON c5#1*.1 (lhtoh); %c0#2% [beta*-2] (mbetal); beta*.5 (vb2); ly1 on alpha1 (balpha2); ly2 on alpha2 (balpha2); ly3 on alpha3 (balpha2); ly4 on alpha4 (balpha2); ly5 on alpha5 (balpha2); ly6 on alpha6 (balpha2); c2#1 ON c1#1@20;! (lmtom); c3#1 ON c2#1@20;! (lmtom); c4#1 ON c3#1@20;! (lmtom); c5#1 ON c4#1@20;! (lmtom); c6#1 ON c5#1@20;! (lmtom); MODEL c0.c1:!specify model for P(Ct C_t-1, C1)!c0 = 1 (high beta); c0 = 2 (low beta) %c0#1.c1#1%!high DV [ly1*2] (inth); ly1*.1 (vqhighb); ly1 ON beta*.1 (bbeta2); c2#1 ON alpha2 (lot11a); %c0#1.c1#2%!low DV [ly1*2] (intl); ly1*.1 (vqhighb); ly1 ON beta*-.1 (bbeta3); c2#1 ON alpha2 (lot21a); 7

%c0#2.c1#1% [ly1*.01] (intm); ly1*1 (vqlowb); ly1 ON beta@0; %c0#2.c1#2% [ly1*.01] (intm); ly1*1 (vqlowb); ly1 ON beta@0; ---------- MODEL c0.c2:!specify model for P(Ct C_t-1, C1) ---------- %c0#1.c2#1% [ly2*2] (inth); ly2*.1 (vqhighb); ly2 ON beta*.1 (bbeta2);!ly2 ON ab2*.5 (bab2); c3#1 ON alpha3 (lot11a); %c0#1.c2#2% [ly2*2] (intl); ly2*.1 (vqhighb); ly2 ON beta*-.1 (bbeta3); c3#1 ON alpha3 (lot21a); %c0#2.c2#1% [ly2*.01] (intm); ly2*1 (vqlowb); ly2 ON beta@0; %c0#2.c2#2% [ly2*.01] (intm); ly2*1 (vqlowb); ly2 ON beta@0; MODEL c0.c3:!specify model for P(Ct C_t-1, C1)! c0 = 1 (high beta); c0 = 2 (low beta); %c0#1.c3#1% [ly3*2] (inth); ly3*.1 (vqhighb); 8

ly3 ON beta*.1 (bbeta2); c4#1 ON alpha4 (lot11a); %c0#1.c3#2% [ly3*2] (intl); ly3*.1 (vqhighb); ly3 ON beta*-.1 (bbeta3); c4#1 ON alpha4 (lot21a); %c0#2.c3#1% [ly3*.01] (intm); ly3*1 (vqlowb); ly3 ON beta@0; %c0#2.c3#2% [ly3*.01] (intm); ly3*1 (vqlowb); ly3 ON beta@0; MODEL c0.c4:!specify model for P(Ct C_t-1, C1)! c0 = 1 (high beta); c0 = 2 (low beta); %c0#1.c4#1% [ly4*2] (inth); ly4*.1 (vqhighb); ly4 ON beta*.1 (bbeta2); c5#1 ON alpha5 (lot11a); %c0#1.c4#2% [ly4*2] (intl); ly4*.1 (vqhighb); ly4 ON beta*-.1 (bbeta3); c5#1 ON alpha5 (lot21a); %c0#2.c4#1% [ly4*.01] (intm); ly4*1 (vqlowb); ly4 ON beta@0; %c0#2.c4#2% [ly4*.01] (intm); ly4*1 (vqlowb); ly4 ON beta@0; 9

MODEL c0.c5:!specify model for P(Ct C_t-1, C1)! c0 = 1 (high beta); c0 = 2 (low beta); %c0#1.c5#1% [ly5*2] (inth); ly5*.1 (vqhighb); ly5 ON beta*.1 (bbeta2); c6#1 ON alpha6 (lot11a); %c0#1.c5#2% [ly5*2] (intl); ly5*.1 (vqhighb); ly5 ON beta*-.1 (bbeta3); c6#1 ON alpha6 (lot21a); %c0#2.c5#1% [ly5*.01] (intm); ly5*1 (vqlowb); ly5 ON beta@0; %c0#2.c5#2% [ly5*.01] (intm); ly5*1 (vqlowb); ly5 ON beta@0; MODEL c0.c6:!specify model for P(Ct C_t-1, C1)! c0 = 1 (high beta); c0 = 2 (low beta); %c0#1.c6#1% [ly6*2] (inth); ly6*.1 (vqhighb); ly6 ON beta*.1 (bbeta2); %c0#1.c6#2% [ly6*2] (intl); ly6*.1 (vqhighb); ly6 ON beta*-.1 (bbeta3); %c0#2.c6#1% [ly6*.01] (intm); ly6*1 (vqlowb); 10

ly6 ON beta@0; %c0#2.c6#2% [ly6*.01] (intm); ly6*1 (vqlowb); ly6 ON beta@0; MODEL CONSTRAINT: New ( MtoOthM sda lowbtom lmtom hiloalp medalp hibeta lobeta inth0 intl0 intm0 hilovy medvy hibtoh0 lowtoh0 lhtoh0 inita_0 lot11a0 lot21a0 HiBp110 HiBp120 HiBp210 HiBp220 HiBp111 HiBp121 HiBp211 HiBp221 HiBp11 HiBp12 HiBp21 HiBp22 dalpha mbetah0 mbetal0 vb10 vb20 lambda1 lambda2); sda = 2.31; MtoOthM = -10; lowbtom = 10; lmtom = 20; hiloalp = balpha1; medalp = balpha2; hibeta = bbeta2; lobeta = bbeta3; inth0 = inth; intl0 = intl; intm0 = intm; hilovy = vqhighb; medvy = vqlowb; hibtoh0 = hibtoh; lowtoh0 = lowtoh; lhtoh0 = lhtoh; inita_0 = inita; lot11a0 = lot11a; lot21a0 = lot21a; mbetah0 = mbetah; 11

mbetal0 = mbetal; vb10 = vb1; vb20 = vb2; lambda1 = l1; lambda2 = l2;!high beta high alpha days HiBp111 = exp(mtoothm + lowtoh + lhtoh + lot11a*sda) /(exp(mtoothm + lowtoh + lhtoh + lot11a*sda) + exp(0)); HiBp121 = exp(0)/ (exp(mtoothm + lowtoh + lhtoh + lot11a*sda) + exp(0)); HiBp211 = exp(mtoothm + lowtoh + lot21a*sda) /(exp(mtoothm + lowtoh + lot21a*sda) + exp(0)); HiBp221 = exp(0)/(exp(mtoothm + lowtoh + lot21a*sda) + exp(0));!high beta low alpha days HiBp110 = exp(mtoothm + lowtoh + lhtoh + lot11a*(-1*sda)) /(exp(mtoothm + lowtoh + lhtoh + lot11a*(-1*sda)) + exp(0)); HiBp120 = exp(0)/ (exp(mtoothm + lowtoh + lhtoh + lot11a*(-1*sda)) + exp(0)); HiBp210 = exp(mtoothm + lowtoh + lot21a*(-1*sda)) /(exp(mtoothm + lowtoh + lot21a*(-1*sda)) + exp(0)); HiBp220 = exp(0)/(exp(mtoothm + lowtoh + lot21a*(-1*sda)) + exp(0));!high beta average alpha days HiBp11 = exp(mtoothm + lowtoh + lhtoh) /(exp(mtoothm + lowtoh + lhtoh) + exp(0)); HiBp12 = exp(0)/ (exp(mtoothm + lowtoh + lhtoh) + exp(0)); HiBp21 = exp(mtoothm + lowtoh) /(exp(mtoothm + lowtoh) + exp(0)); HiBp22 = exp(0)/(exp(mtoothm + lowtoh) + exp(0)); inth > intm; intm > intl; balpha2 = balpha1 + dalpha; 12

!-------------------------------------------------- OUTPUT: TECH1 TECH4 CINTERVAL; SAVEDATA: RESULTS = 01outPar3.txt ; FILE = 01LTA3.class ; RECORDLENGTH IS 5000; SAVE = CPROBABILITIES; 13

0.3 Tables of Simulation Results The tables included here show the summary statistics of the parameter estimates from all remaining MSEM and MSEM-RS models considered but not presented in Chow et al. s the paper entitled The Cusp Catastrophe Model as Cross-Sectional and Longitudinal Mixture Structural Equation Models. Table 1: Summary Statistics of Parameter Estimates for MSEM Model 2 with T = 1 and n = 500 across 500 Monte Carlo Replications. Mean ˆθ SD 2.5 %tile 97.5%tile aŝe τ high 1.10 0.07 0.97 1.23 0.07 b α,high&low 0.12 0.01 0.09 0.14 0.01 b β,high 0.19 0.01 0.17 0.21 0.01 ψ high&low 0.11 0.01 0.09 0.14 0.01 τ low -1.10 0.07-1.25-0.97 0.07 b β,low -0.19 0.01-0.21-0.17 0.01 τ med 0.01 0.12-0.21 0.25 0.11 b α,med 0.18 0.07 0.02 0.27 0.05 ψ med 0.18 0.06 0.05 0.30 0.05 a 10, logit intercept for R high β -0.00 0.09-0.17 0.16 0.09 b 111,, LO(high β high) 2.52 2.52 0.17 7.12 1.51 b 121,, LO(high β low) 2.37 2.10 0.08 6.54 1.36 a 11, logit intercept for LO(low β high) -1.59 2.49-5.70 0.31 1.32 a 21, logit intercept for LO(low β low) -1.43 2.09-5.56 0.58 1.17 b 111,α, α LO(high β high) 1.70 1.18 0.48 4.36 0.67 b 121,α, α LO(high β low) -1.74 1.53-3.91-0.47 0.68 b 211,α, α LO(low β high) 0.89 0.84 0.17 3.15 0.53 b 221,α, α LO(low β low) -0.88 0.73-2.79-0.19 0.48 Pr(high β high) 0.41 0.10 0.20 0.60 0.10 Pr(high β low) 0.41 0.10 0.22 0.59 0.10 Pr(high β med) 0.18 0.08 0.03 0.36 0.08 Pr(low β high) 0.18 0.11 0.00 0.42 0.10 Pr(low β low) 0.19 0.12 0.00 0.45 0.10 Pr(low β med) 0.63 0.17 0.26 0.91 0.14 a The class and transition probabilities were computed using Equations 8, 9 and 12 with α i set to 0. Proportion(cases free of SE estimation problems for logit parameters in Model 2) =.94 14

Table 2: Summary Statistics of Parameter Estimates for MSEM Model 2 with T = 1 and n = 200 across 500 Monte Carlo Replications. Mean ˆθ SD 2.5 %tile 97.5%tile aŝe τ high 1.10 0.10 0.90 1.29 0.09 b α,high&low 0.11 0.02 0.07 0.16 0.02 b β,high 0.19 0.02 0.17 0.23 0.01 ψ high&low 0.11 0.02 0.07 0.15 0.02 τ low -1.10 0.11-1.32-0.92 0.09 b β,low -0.19 0.02-0.23-0.16 0.02 τ med 0.01 0.30-0.63 0.95 0.13 b α,med 0.18 0.14-0.06 0.39 0.05 ψ med 0.13 0.08 0.00 0.30 0.05 a 10, logit intercept for R high β 0.01 0.14-0.30 0.26 0.14 b 111,, LO(high β high) 3.31 6.92-3.55 26.31 2.59 b 121,, LO(high β low) 3.17 7.01-3.36 24.60 2.48 a 11, logit intercept for LO(low β high) -1.17 5.42-12.82 5.74 2.13 a 21, logit intercept for LO(low β low) -1.13 5.39-12.05 4.97 2.02 b 111,α, α LO(high β high) 1.95 2.28-0.39 6.28 0.95 b 121,α, α LO(high β low) -2.29 2.57-9.23-0.04 0.93 b 211,α, α LO(low β high) 1.11 1.75-0.41 5.53 0.85 b 221,α, α LO(low β low) -1.01 1.59-5.57 0.36 0.68 Pr(high β high) a 0.42 0.17 0.12 0.80 0.14 Pr(high β low) 0.40 0.17 0.03 0.74 0.14 Pr(high β med) 0.18 0.13 0.00 0.48 0.10 Pr(low β high) 0.23 0.18 0.00 0.69 0.13 Pr(low β low) 0.24 0.18 0.00 0.65 0.13 Pr(low β med) 0.52 0.26 0.00 0.95 0.17 a The class and transition probabilities were computed using Equations 8, 9 and 12 with α i set to 0. Proportion(cases free of SE estimation problems for logit parameters in Model 2) =.54 15

Table 3: Summary Statistics of Parameter Estimates for MSEM Model 6 with T = 1 and n = 200 across 500 Monte Carlo Replications. True θ Mean ˆθ SD 2.5 %tile 97.5%tile aŝe λ β,1 1.20 1.20 0.01 1.17 1.23 0.01 λ β,2 0.90 0.90 0.01 0.88 0.92 0.01 ψɛ,β 2 1 0.09 0.09 0.01 0.07 0.12 0.01 ψɛ,β 2 2 0.09 0.09 0.02 0.06 0.13 0.02 ψɛ,β 2 3 0.09 0.09 0.01 0.07 0.11 0.01 µ β,high 2.50 2.50 0.07 2.36 2.64 0.07 ψ β,high 0.49 0.47 0.07 0.33 0.63 0.07 µ β,low -2.50-2.50 0.08-2.64-2.36 0.07 ψ β,low 0.49 0.49 0.07 0.37 0.63 0.07 τ high 1.05 0.15 0.69 1.32 0.09 b α,high&low 0.12 0.02 0.08 0.17 0.02 b β,high 0.20 0.05 0.12 0.30 0.02 ψ high&low 0.11 0.02 0.08 0.16 0.02 τ low -1.01 0.17-1.29-0.62 0.09 b β,low -0.21 0.06-0.38-0.13 0.02 τ med 0.02 0.41-0.77 0.88 0.13 b α,med 0.17 0.14-0.15 0.39 0.04 ψ med 0.12 0.08 0.00 0.28 0.04 a 10, logit intercept for R high β 0.00 0.00 0.01-0.01 0.02 0.14 b 111,, LO(high β high) 2.66 6.97-8.94 25.84 5.32 b 121,, LO(high β low) 2.91 7.41-8.82 25.46 2.70 a 11, logit intercept for LO(low β high) -1.23 5.64-15.58 8.32 4.91 a 21, logit intercept for LO(low β low) -1.38 6.57-23.28 9.17 2.40 b 111,α, α LO(high β high) 2.10 2.36-0.00 7.18 0.97 b 121,α, α LO(high β low) -1.95 1.89-5.48-0.03 0.90 b 211,α, α LO(low β high) 1.10 2.02-2.45 6.85 1.39 b 221,α, α LO(low β low) -0.70 2.47-4.27 2.47 0.83 Pr(high β high) a 0.38 0.17 0.07 0.71 0.15 Pr(high β low) 0.41 0.17 0.08 0.74 0.15 Pr(high β med) 0.20 0.14 0.00 0.53 0.11 Pr(low β high) 0.23 0.18 0.00 0.61 0.13 Pr(low β low) 0.30 0.24 0.00 0.81 0.13 Pr(low β med) 0.46 0.30 0.00 0.96 0.16 a The class and transition probabilities were computed using Equations 8, 9 and 12 with α i set to 0. Proportion(cases free of SE estimation problems for logit parameters in Model 2) =.38 16

Table 4: Summary Statistics of Parameter Estimates for MSEM-RS Model 1 with T = 6 and n = 500 across 500 Monte Carlo Replications. τ high 0.59 0.06 0.48 0.71 0.06 b α,high&low 0.24 0.00 0.23 0.25 0.01 b β,high 0.27 0.02 0.22 0.31 0.02 ψ high&low 0.18 0.01 0.17 0.20 0.01 τ low -0.59 0.06-0.71-0.46 0.06 b β,low -0.27 0.02-0.32-0.22 0.02 τ med -0.00 0.01-0.03 0.03 0.01 b α,med 0.24 0.00 0.23 0.25 0.01 ψ med 0.26 0.01 0.24 0.28 0.01 b 111,, LO(high β high) -10.01 0.31-10.66-9.39 0.30 a 1,1t,, LO(low high high β) 10.00 0.20 9.61 10.39 0.19 b 1,11t,, LO(high high high β) 0.00 0.27-0.57 0.55 0.27 b 111,α, α LO(high β high) 2.66 0.53 1.94 3.93 0.47 b 1,11t,α, α LO(high high high β) 2.57 0.32 2.06 3.32 0.29 b 1,21t,α, α LO(low high high β) 2.57 0.32 2.08 3.25 0.29 Pr(high high high β, low α) a 0.00 0.00 0.00 0.01 0.00 Pr(high low high β, low α) 1.00 0.00 0.99 1.00 0.00 Pr(low high high β, low α) 0.00 0.00 0.00 0.01 0.00 Pr(low low high β, low α) 1.00 0.00 0.99 1.00 0.00 Pr(high high high β, high α) 1.00 0.00 0.99 1.00 0.00 Pr(high low high β, high α) 0.00 0.00 0.00 0.01 0.00 Pr(low high high β, high α) 1.00 0.00 0.99 1.00 0.00 Pr(low low high β, high α) 0.00 0.00 0.00 0.01 0.00 Pr(high high high β, avg α) 0.50 0.05 0.41 0.59 0.05 Pr(high low high β, avg α) 0.50 0.05 0.41 0.59 0.05 Pr(low high high β, avg α) 0.50 0.05 0.40 0.60 0.05 Pr(low low high β, avg α) 0.50 0.05 0.40 0.60 0.05 a The transition probabilities were computed using Equations 13 and 14. Avg α = value of α was set to 0; high α = value of α was set to 1 SD above the mean of 0; low α = value of α was set to 1 SD below the mean of 0. Proportion(cases free of SE estimation problems for logit parameters in Model 2) = 1.00 17

Table 5: Summary Statistics of Parameter Estimates for MSEM-RS Model 1 with T = 6 and n = 200 across 500 Monte Carlo Replications. τ high 0.59 0.10 0.38 0.80 0.04 b α,high&low 0.24 0.01 0.23 0.26 0.00 b β,high 0.27 0.04 0.19 0.34 0.01 ψ high&low 0.18 0.01 0.16 0.21 0.01 τ low -0.59 0.11-0.79-0.39 0.03 b β,low -0.27 0.04-0.34-0.19 0.00 τ med 0.00 0.02-0.04 0.04 0.02 b α,med 0.24 0.01 0.23 0.26 0.00 ψ med 0.26 0.01 0.24 0.29 0.02 b 111,, LO(high β high) -10.04 0.58-11.34-8.88 0.52 a 1,1t,, LO(low high high β) 10.01 0.30 9.43 10.55 0.30 b 1,11t,, LO(high high high β) 0.01 0.43-0.91 0.82 0.43 b 111,α, α LO(high β high) 2.94 1.23 1.61 6.12 0.79 b 1,11t,α, α LO(high high high β) 2.62 0.51 1.88 3.75 0.46 b 1,21t,α, α LO(low high high β) 2.62 0.52 1.87 3.95 0.46 Pr(high high high β, low α) 0.00 0.00 0.00 0.01 0.00 Pr(high low high β, low α) 1.00 0.00 0.99 1.00 0.00 Pr(low high high β, low α) 0.00 0.00 0.00 0.01 0.00 Pr(low low high β, low α) 1.00 0.00 0.99 1.00 0.00 Pr(high high high β, high α) 1.00 0.00 0.99 1.00 0.00 Pr(high low high β, high α) 0.00 0.00 0.00 0.01 0.00 Pr(low high high β, high α) 1.00 0.00 0.99 1.00 0.00 Pr(low low high β, high α) 0.00 0.00 0.00 0.01 0.00 Pr(high high high β, avg α) 0.50 0.07 0.35 0.64 0.07 Pr(high low high β, avg α) 0.50 0.07 0.36 0.65 0.07 Pr(low high high β, avg α) 0.50 0.07 0.36 0.63 0.07 Pr(low low high β, avg α) 0.50 0.07 0.37 0.64 0.07 a The transition probabilities were computed using Equations 13 and 14. Avg α = value of α was set to 0; high α = value of α was set to 1 SD above the mean of 0; low α = value of α was set to 1 SD below the mean of 0. Proportion(cases free of SE estimation problems for logit parameters in Model 2) =.93 18

Table 6: Summary Statistics of Parameter Estimates for MSEM-RS Model 5 with T = 6 and n = 200 across 500 Monte Carlo Replications. True θ Mean ˆθ SD 2.5 %tile 97.5%tile aŝe λ β,1 1.20 1.20 0.01 1.17 1.22 0.01 λ β,2 0.90 0.90 0.01 0.88 0.92 0.01 ψɛ,β 2 1 0.09 0.09 0.01 0.07 0.12 0.01 ψɛ,β 2 2 0.09 0.09 0.02 0.06 0.12 0.02 ψɛ,β 2 3 0.09 0.09 0.01 0.07 0.11 0.01 µ β,high 2.50 2.50 0.07 2.35 2.64 0.07 ψ β,high 0.49 0.49 0.09 0.34 0.66 0.08 µ β,low -2.50-2.50 0.07-2.64-2.35 0.07 ψ β,low 0.49 0.48 0.07 0.36 0.62 0.07 τ high 0.54 0.20 0.00 0.78 0.14 b α,high&low 0.25 0.01 0.23 0.26 0.03 b β,high 0.28 0.09 0.20 0.49 0.04 ψ high&low 0.18 0.01 0.16 0.21 0.02 τ low -0.50 0.24-0.79 0.00 0.12 b β,low -0.29 0.10-0.49-0.19 0.04 τ med -0.00 0.02-0.05 0.04 0.02 b α,med 0.25 0.01 0.23 0.26 0.03 ψ med 0.26 0.01 0.23 0.29 0.02 a 10, logit intercept for R high β 0.00 0.00 0.01-0.00 0.02 0.14 b 111,, LO(high β high) -9.99 0.95-11.56-8.72 0.58 a 1,1t,, LO(low high high β) 10.01 0.33 9.37 10.63 0.31 b 11t,, LO(high high high β) -0.01 0.45-0.84 0.82 0.45 b 111,α, α LO(high β high) 3.18 2.14 1.60 7.37 0.95 b 11t,α, α LO(high high) 2.64 0.61 1.93 3.81 0.50 b 21t,α, α LO(low high) 2.69 0.65 1.88 4.00 0.50 Pr(high high high β, low α) a 0.01 0.06 0.00 0.01 0.00 Pr(high low high β, low α) 0.99 0.06 0.99 1.00 0.00 Pr(low high high β, low α) 0.01 0.06 0.00 0.01 0.00 Pr(low low high β, low α) 0.99 0.06 0.99 1.00 0.00 Pr(high high high β, high α) 0.99 0.06 0.99 1.00 0.00 Pr(high low high β, high α) 0.01 0.06 0.00 0.01 0.00 Pr(low high high β, high α) 0.99 0.06 0.99 1.00 0.00 Pr(low low high β, high α) 0.01 0.06 0.00 0.01 0.00 Pr(high high high β, avg α) 0.50 0.08 0.35 0.63 0.08 Pr(high low high β, avg α) 0.50 0.08 0.37 0.65 0.08 Pr(low high high β, avg α) 0.50 0.08 0.35 0.65 0.08 Pr(low low high β, avg α) 0.50 0.08 0.35 0.65 0.08 a The transition probabilities were computed using Equations 13 and 14. Avg α = value of α was set to 0; high α = value of α was set to 1 SD above the mean of 0; low α = value of α was set to 1 SD below the mean of 0. Proportion(cases free of SE estimation problems for logit parameters in Model 2) =.99 19